Question

interigate 1/2x+3.......,......​

Answer 1

Answer:

(1/2) . log (2x - 3) + C, where C = constant of integration

Proof:

Let I = ∫ dx/(2x - 3) …..………………………………………….(1)

Here x is the independent variable. To evaluate the integral in (1), we take recourse to the substitution method.

Put 2x - 3 = y ………………………………….…………………(2)

Taking differentials,

2. dx - 0 = dy

⇒ dx = dy/2

Substituting for dx and (2x-3) in (1),

I = ∫dy/2y = (1/2) ∫dy/y = (1/2) . log y + C

Now revert to the original variable x using (2).

∴ I = (1/2) . log (2x - 3) + C where C = constant of integration.

Hence integral of 1/(2x - 3) dx = [log (2x - 3)]/2 + C (Answer)

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